I own a newtonian reflector and occasionally use it (using the reflection method with paper) for looking at the sun. I happen to also own a refractor, but I'm not exactly sure if the reflection method will damage the equipment (notably the lens on the refractor) if I do.

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Actually, your Newtonian may run the risk of suffering from the projection method. The secondary, planar mirror in the Newtonian will be subject to heating. Although it is not in the prime focal point, the light rays are already pretty much bundled up, and this may lead to heating and ultimately to failure of the secondary mirror!

With refractors, you don't have anything in the optical path up until the eyepiece, so in a sense it is more safe.

However: I strongly suggest against using simple projection methods. One alternative is to use a Herschel-wedge. However, these are again not that useful on Newtonians, because they filter the light at the eyepiece end of the telescope. It is probably better to buy a solar filter which is rather inexpensive and can be fitted to any kind of telescope. Without wanting to promote any specific brand, the company Baader has a good explanation on the topic on their website. The important thing is that the filter is securely fastened to the optical tube assembly, and that there are no scratches or cuts in the filter foil.

Another much more expensive method is to use a specialized solar telescope. These usually use very narrowband filters and coatings on the telescope lenses. Often they are specialized to one of the hydrogen or calcium spectral lines, showing great amount of structure in the suns atmosphere.